PEAR: a Tool for Reasoning About Scenarios and Probabilities in Description Logics of Typicality (short paper)

PEAR: a Tool for Reasoning About Scenarios and Probabilities in Description Logics of

Typicality

Gian Luca Pozzato^1? and Gabriele Soriano¹ Dipartimento di Informatica, Universit`a di Torino, Italy - gianluca.pozzato@unito.it,gabriele.soriano@edu.unito.it

Abstract. In this work we describe PEAR, a tool for reasoning about prototypical properties in an extension of Description Logics of typical- ity with probabilities and scenarios. PEAR implements a non-monotonic procedure for the logicALC+T^P_R, a recently introduced extension of the logic of typicalityALC+TRby inclusions of the formT(C)vpD, where pis a real number between 0 and 1 capturing the intuition that “all the typicalCs areDs, and the probability that aC is not aDis 1−p”. In this logic, different scenarios are considered by taking into account sev- eral extension of the ABox, containing only some typicality assumptions about individuals. Each scenario has a probability depending on those equipping typicality inclusions, then entailment can be restricted to sce- narios whose probabilities belong to a given and fixed range. PEAR is implemented in Python, it computes all scenarios of a knowledge base and it allows the user to check the probability of a query by exploiting a translation into standardALC.

1 Introduction

Non-monotonic extensions of Description Logics (from now on, DLs for short) have been actively investigated since the early 90s [1–6], in order to tackle the well known problem of allowing one to represent and reason about prototypi- cal properties and defeasible inheritance. A simple but powerful non-monotonic extension of DLs is proposed in [7]: in this approach, typical properties can be directly specified by means of a “typicality” operator: a TBox, in addition to standard inclusionsCvD, representing that “allCs are alsoDs”, can contain inclusions of the form T(C) v D to represent that “typical Cs are also Ds”.

The Description Logic so obtained is calledALC+TRand, as a difference with standard DLs, one can consistently express exceptions and reason about defeasi- ble inheritance as well. For instance, a knowledge base can consistently contain the inclusions (1) HummingBird v Bird, (2) T(Bird)v ¬BackwardsFlier and (3) T(HummingBird) v BackwardsFlier, expressing that, normally, birds are not able to fly backwards, however hummingbirds are exceptional birds that, typically, are able to fly backwards. The key point of the logicALC+TR relies

?The author is partially supported by the project “METALLIC #2: METodi di prova per il ragionamento Automatico per Logiche non-cLassIChe #2”, INdAM GNCS 2019.

on the fact that the semantics of the T operator is strongly related to axioms and rules of rational entailment as introduced in [8], widely considered as the core properties of non-monotonic reasoning. As a consequence,Tinherits well- established properties like specificity: in the example, if one knows that Petey is a typical hummingbird, then the logic ALC +TR allows one to infer that it flies backwards, giving preference to the most specific information of being a hummingbird with respect to being a bird.

The main drawback of the logic ALC +T_R it that it is too weak: in- deed, although the operator T is non-monotonic (T(C) v E does not imply T(C uD) v E), the logic ALC +TR is monotonic [9, 10], namely, if F is entailed from a knowledge base KB, then F is also entailed from any KB’ ⊇ KB. As a consequence, unless a KB contains explicit assumptions about typ- icality of individuals, there is no way of inferring defeasible properties about them: in the above example, if KB only contains that Petey is a hummingbird (HummingBird(petey)), it is not possible to conclude that it flies backwards:

this would be possible only if the stronger information that Petey is a typical hummingbird (T(HummingBird)(petey)) belongs to (or can be inferred from) KB. In order to tackle this problem, in [9, 11] the authors have strengthened the semantics of the logicALC+T_Rby means of a minimal model semantics, corre- sponding to a notion ofrational closure as defined in [8] for propositional logic.

Intuitively, the idea is to restrict reasoning to (canonical) models that maximize typical instances of a concept when consistent with the knowledge base. As a consequence, in the resulting logic ALC+T^RaCl_R , if one knows that Petey is a hummingbird, one can nonmonotonically assume that it is also atypical one if this is consistent, and therefore that it is able to fly backwards. From a semantic point of view, the logic ALC+T^RaCl_R is based on a preference relation among ALC+TRmodels and a notion ofminimal entailmentrestricted to models that are minimal with respect to such preference relation.

The logicALC+T^RaCl_R imposes to considerall typicality assumptions that are consistent with a KB. In other words, in absence of explicit contradictory information, the logicALC+T^RaCl_R leads to assume that there are no exceptions:

this seems to be too strong in several application domains, especially in situations characterized by a high number of individuals that are all considered as typical ones, whereas it could be useful to reason about scenarios exhibitingexceptional ones. To this aim, in [12] we have introduced a further extension of the logic of typicality calledALC+T^P_R, which extendsALCby means of typicality inclusions equipped by probabilities of the form T(C) vp D, where p ∈ (0,1), whose intuitive meaning is that “normally, Cs are Ds and the probability of having exceptional Cs – not being Ds – is 1−p”. All the typical instances of the conceptC are also instances of the concept D, but we have the opportunity of expressing a degree about the fact that a C element is not also a D element, i.e. it is an exceptional C element. This allows us to rank different typicality inclusions, for instance we can have thatT(TeenAger)v_0.6InfluencerFollower as well as thatT(TeenAger)v0.9∃play.VideoGame, capturing the intuition that both following an influencer and playing video games are typical properties of

teenagers, and we also want to express that the probability of having exceptional teenagers not following any influencer is higher than the one of finding ones not playing video games, by ranking the two properties with probabilities 0.6 and 0.9, respectively.

In this work we introduce PEAR, a preliminary implementation of the rea- soning mechanism for the logicALC+T^P_Rdescribed in [12]. PEAR, standing for Probability of Exceptions and typicAlity Reasoner, computes different syntactic extensions of an ABox containing only some of the “plausible” typicality asser- tions that can be entailed from the KB in the non-monotonic logicALC+T^RaCl_R : each extension represents ascenario having a specific probability. Then, PEAR allows one to check whether a query is non-monotonically entailed from the knowledge base, either by selecting the scenarios where it holds or by restricting the attention to extensions whose probabilities belong to a given and fixed range, as well as to compute the probability of a query as the sum of the probabilities of scenarios from which it is entailed. This allows one to reason about scenarios that are not necessarily the most probable and are, in some sense, surprising.

The decision procedure computed by PEAR isExpTimecomplete, therefore we retain the same complexity of the underlying standardALC.

2 Preferential Description Logics

Let us first quickly recall the main notions about the Description Logic of typ- icality ALC+TR introduced in [7, 13, 9]. The logic ALC+TR is obtained by adding to standardALCthe typicality operatorT[7]. The intuitive idea is that T(C) selects thetypicalinstances of a conceptC. We can therefore distinguish between the properties that hold for all instances of concept C (C vD), and those that only hold for the normal or typical instances of C (T(C) v D).

From a semantic point of view, we refer to rational models [9]: a model M is any structureh∆^I, <, .^Iiwhere∆^Iis the domain,<is an irreflexive, transitive, well-founded and modular (for all x, y, z in ∆^I, if x < y then either x < z or z < y) relation over∆^I. In this respect, x < ymeans that xis “more normal”

thany, and that the typical members of a conceptC are the minimal elements of C with respect to this relation. An element x∈ ∆^I is a typical instance of some concept C if x∈ C^I and there is no C-element in ∆^I more typical than x. In detail,.^I is the extension function that maps each conceptCtoC^I⊆∆^I, and each roleRtoR^I⊆∆^I×∆^I. For concepts ofALC,C^Iis defined as usual.

For theToperator, let M in<(C^I) ={x∈C^I |@y∈C^I s.t.y < x}, we define (T(C))^I=M in<(C^I).

A model M can be equivalently defined by postulating the existence of a functionk_M:∆^I7−→N, wherek_Massigns a finite rank to each domain element:

the rank functionk_Mand<can be defined from each other by stating thatx < y if and only if k_M(x)< k_M(y). Given standard definitions of satisfiability of a KB in a model, we define a notion of entailment inALC+T_R. Given a query F (either an inclusion C vD or an assertionC(a) or an assertion of the form R(a, b)), we say thatF is entailed from a KB ifF holds in allALC+TR models satisfying KB.

As already mentioned in the Introduction, even if theToperator itself is non- monotonic, the logic ALC+T_R is monotonic. In order to perform useful non- monotonic inferences, in [9] the authors have strengthened the above semantics by restricting entailment to a class of minimal models. Intuitively, the idea is to restrict entailment to models thatminimize the untypical instances of a concept.

The resulting logic is called ALC+T^RaCl_R and it corresponds to a notion of rational closure on top of ALC+TR. Such a notion is a natural extension of the rational closure construction provided in [8] for the propositional logic. The non-monotonic semantics ofALC+T^RaCl_R relies on minimal rational models that minimize therank of domain elements. Informally, given two models of KB, one in which a given domain elementxhas rank 2 (because for instancez < y < x), and another in which it has rank 1 (because onlyy < x), we prefer the latter, as in this model the elementxis assumed to be “more typical” than in the former.

Query entailment is then restricted to minimalcanonical models. The intuition is that a canonical model contains all the individuals that enjoy properties that are consistent with KB. A modelMis a minimal canonical model of KB if it satisfies KB, it is minimal and it is canonical¹. A queryF is minimally entailed from a KB if it holds in all minimal canonical models of KB. In order to ascribe typical properties to individuals, the notion of rational closure is further extended to the ABox. In [9] it is shown that query entailment inALC+T^RaCl_R is inExpTime.

3 The Logic ALC + T

In this section we recall the logic ALC+T^P_R, extending the DLs of typicality with probabilities and scenarios. Here, typicality inclusion are equipped with a probability/degree of not having exceptions for it, and have the form

T(C)v_pD,

whose meaning is “typical Cs are alsoDs, and the probability of having excep- tionalCs not beingDs is 1−p”.

Definition 1. We consider an alphabet of concept names C, of role names R, and of individual constants O. GivenA∈ C andR∈ R, we define:

C:=A| > | ⊥ | ¬C|CuC|CtC| ∀R.C| ∃R.C

An ALC+T^P_R knowledge base is a pair (T,A).T contains axioms of the form either C vC or T(C)vp C, where p∈R, p∈(0,1). Acontains assertions of the form either C(a)or R(a, b), wherea, b∈ O.

Given an inclusionT(C)vpD, the higher the probabilityp, the less the proba- bility of having exceptionalCs not being alsoDs. In this respect, the probability pis a real number included in the open interval (0,1): the probability 1 is not allowed, in the sense that an inclusionT(C)v₁D would correspond to astrict

1 In Theorem 10 in [9] the authors have shown that for any consistent KB there exists a finite minimal canonical model of KB.

inclusion C vD (all Cs are Ds). Given another inclusion T(C⁰)v_p⁰ D⁰, with p⁰< p, we assume that this inclusion is less “strict” than the other one, i.e. the probability of having exceptionalC⁰s is higher than the one of having exceptional Cs with respect to propertiesD⁰ andD, respectively.

Given a KB, we define the finite setTipof concepts occurring in the scope of the typicality operator, i.e.Tip={C|T(C)vp D∈ KB}. Given an individual a explicitly named in the ABox, we define the set of typicality assumptions T(C)(a) that can be minimally entailed from KB in the non-monotonic logic ALC+T^RaCl_R , with C ∈ Tip. We then consider an ordered set Tip_A of pairs (a, C) of all possible assumptions T(C)(a), for all concepts C ∈ Tip and all individual constantsain the ABox. We then define the ordered multiset PA= [p1, p2, . . . , pn], respecting the order imposed onTip_A, wherepi=

m

Q

j=1

pij for all T(Ci)vp_i1 D1,T(Ci)vp_i2 D2, . . . ,T(Ci)vp_im Dm inT. The ordered multiset P_A is a tuple of the form [p1, p2, . . . , pn], where pi is the probability of the assumptionT(C)(a), such that (a, C)∈Tip_A at positioni.pi is the product of all the probabilitiespij of inclusions T(C)vp_ij D in the TBox.

Following the basic idea underlying surprising scenarios outlined in [14], we consider different extensionsAfiof the ABox and we equip them with a probabil- ityPi. Starting fromPA= [p1, p2, . . . , pn], the first step is to build all alternative tuples where 0 is used in place of some pi to represent that the corresponding typicality assertionT(C)(a) is no longer assumed. We then define the notion of extension of the ABox corresponding to a string so obtained: in this way, the extension corresponding toPA contain all the available typicality assumptions;

in the other extensions, some typicality assumptions are discarded, thus 0 is used in place of the correspondingp_i. The probability of an extensionAf_i correspond- ing to a string P_Ai= [p_i1, p_i2, . . . , p_in] is defined as the product of probabilities p_ij when p_ij 6= 0, i.e. the probability of the corresponding typicality assump- tion when this is selected for the extension, and 1−pj when pij = 0, i.e. the corresponding typicality assumption is discarded, that is to say the extension contains an exception to the inclusion. It is easy to observe that we obtain a probability distribution over extensions ofA.

Non-monotonic entailment of a queryF in the Description LogicALC+T^P_R is defined as follows: (i) ifF is a TBox inclusionCvD, then it is entailed from KB if it is minimally entailed from KB⁰ in the non-monotonic ALC +T^RaCl_R , where KB⁰ is obtained from KB by removing probabilities of exceptions, i.e. by replacing each typicality inclusion T(C) vp D withT(C)v D; (ii) if F is an ABox fact C(a), then it is entailed from KB if it is entailed in the monotonic ALC+T_Rfrom scenarios extending KB withAi. Furthermore, one can consider a notion of entailment restricted to scenarios whose probabilities belong to a given range and, similarly to [15], a notion of probability of the entailment of a queryC(a), as the sum of the probabilities of all extensions from whichC(a) is so entailed. In [12] it is shown that entailment inALC+T^P_RisExpTime-complete as for standardALC.

4 Design of PEAR

In this section we introduce PEAR (Probability of Exceptions and typicAlity Reasoner), a Python implementation of the reasoning services provided by the logicALC+T^P_R. PEAR makes use of the libraryowlready2²that allows one to rely on the services of efficient DL reasoners, e.g. the HermiT reasoner. PEAR exploits the translation of an ALC+TR knowledge base into standard ALC introduced in [9] and described later in this section. The current version of the system PEAR, along with the files for the examples presented in this paper, are available at http://di.unito.it/pear.

The system PEAR implements a non-monotonic procedure whose aim is to compute extensionsA of the ABox and corresponding alternative scenarios having different probabilities described in the previous section. Given anALC+ T^P_R KB=(T,A) and a query F, PEAR computes the following four steps: 1.

compute the set Tip_a of all typicality assumptions that are minimally entailed from the knowledge base in the non-monotonic logicALC+T^RaCl_R ; 2. compute all possiblefA_iextensions of the ABox and compute their probabilities; 3. select the extensions whose probabilities belong to a given range hp, qi; 4. check whether the query F is entailed from all the selected extensions in the monotonic logic ALC+TR. Step 1 is based on reasoning in the non-monotonic logicALC+T^RaCl_R : in this case, PEAR computes the rational closure of an ALC+TR knowledge base by means of the algorithm introduced in [9], which is sound and complete with respect to the minimal model semantics recalled in Section 2. Step 4 is based on reasoning in the monotonic logicALC+TR: to this aim, the procedure relies on a polynomial encoding ofALC+TRintoALCintroduced in [16], which exploits the definition ofTin terms of a G¨odel-L¨ob modality2as follows:T(C) is defined as Cu2¬C where the accessibility relation of the modality2 is the preference relation<inALC+TRmodels. Let KB=(T,A) be a knowledge base whereAdoes not contain positive typicality assertions on individuals of the form T(C)(a). The encoding KB’=(T⁰,A⁰) of KB inALC is defined as follows. First of all, we letA⁰=∅. Then, for eachAvB∈ T, not containingT, we introduce AvB inT⁰. For eachT(A) occurring inT, we introduce a new atomic concept Box_¬A and, for each inclusion T(A) v_p B ∈ T, we add to T⁰ the inclusion AuBox_¬A vB. In order to capture the properties of the 2 modality, a new roleR is introduced to represent the relation <in preferential models, and the following inclusions are introduced in T⁰: (i) Box_¬A v ∀R.(¬AuBox_¬A) and (ii)¬Box_¬Av ∃R.(AuBox_¬A). Inclusion (i) accounts for the transitivity of<, whereas inclusion (ii) accounts for the well-foundedness, namely the fact that if an element is not a typicalA element then there must be a typical A element preferred to it. For the encoding of the inclusions, if ClvCris not a typicality inclusion, then C_l⁰ = Cl and C_r⁰ = Cr; if Cl v Cr is a typicality inclusion T(A)vCr, thenC_l⁰=AuBox¬A andC_r⁰ =Cr. The size of KB’ is polynomial in the size of the KB. The same forC_l⁰ and C_r⁰, assuming the size ofCland Cr

be polynomial in the size of KB. As an example, let the TBox contain:

2 https://pythonhosted.org/Owlready2/

gives information to

computed by

manage order of execution computed by

manage information by USER

Reasoning and query files

Auxiliary file Text file

Fig. 1.The main components of the tool PEAR.

HummingBirdvBird T(Bird)v ¬BackwardsFlier

T(HummingBird)vBackwardsFlier The system PEAR builds KB’ containing:

HummingBirdvBird

BirduBird1 v ¬BackwardsFlier Bird1 v ∀R1.(¬BirduBird1)

¬Bird1 v ∃R1.(BirduBird1)

HummingBirduHummingBird1 vBackwardsFlier

HummingBird1 v ∀R2.(¬HummingBirduHummingBird1)

¬HummingBird1 v ∃R2.(HummingBirduHummingBird1)

The system PEAR comprises eleven Python files, whose behaviour and de- pendencies are summarized in Figure 4. In particular, the following modules represent the core components of the tool:

– OntologyManager.py, whose aim is to manage the knowledge base imple- menting the above translation into standardALC;

– IncreasedOntology.py, whose objectives are to generate all scenarios and to compute their probabilities;

– ReasoningOnScenarios.py, which is devoted to check entailment of a query (QueryInput.py) by exploiting reasoning services provided by HermiT.

Let us explain the functioning of PEAR by an example inspired by [12]. Let a KB be a knowledge base inALC+T^P_R, where the TBox is as follows:

PokemonCardPlayer vCardPlayer (1) T(CardPlayer)v_0.85¬YoungPerson (2) T(PokemonCardPlayer)v_0.7YoungPerson (3) T(Student)v0.6YoungPerson (4) T(Student)v0.8InstagramUser (5) whereas the ABox is{PokemonCardPlayer(lollo),Student(thomas)}. PEAR first computes the set of typicality assumptions entailed fromALC+T^RaCl_R , namely (a) T(PokemonCardPlayer)(lollo) and (b) T(Student)(thomas). Then, PEAR generates the following four different scenarios:

1. neither (a) nor (b) are assumed: the probability is (1−0.7)×(1−(0.6×0.8)) = 0.156, but PEAR is not able to conclude anything about Lollo and Thomas;

2. (a) is assumed, whereas (b) is not: this scenario, having a probability 0.7× (1−(0.6×0.8)) = 0.364, allows PEAR to concludeYoungPerson(lollo);

3. (b) is assumed, whereas (a) is not: this scenario has probability (1−0.7)× (0.6×0.8) = 0.144, and PEAR only concludes YoungPerson(thomas) and InstagramUser(thomas);

4. both (a) and (b) are assumed: in this scenario, whose probability is 0.7× (0.6×0.8) = 0.336. In this scenario, PEAR concludes that both Lollo and Thomas are young persons and that Thomas is an Instagram user.

PEAR can also evaluates the probability of a query as the sum of the proba- bilities of scenarios where such a query holds. In the example, PEAR evaluates the probability that Lollo is a young person as the sum of the probabilities of scenarios (2) and (4), and it is therefore 0.336 + 0.364 = 0.7. Similarly, the prob- ability that Thomas is an Instagram user is 0.336 + 0.144 = 0.48. Some pictures of the execution of PEAR in this example are provided in Figure 4.

5 Related Works and Conclusions

In this work we have introduced PEAR, a tool for reasoning in the non-monotonic Description LogicALC+T^P_R, which extends the logic of typicality by means of probabilities equipping typicality inclusions. In this setting, T(C)vp D is in- tended as “normally, Cs are Ds and we have a probability of 1−p of having exceptional Cs not being Ds”. From a knowledge representation point of view, as a difference fromALC+T^RaCl_R , the logicALC+T^P_Rallows one to distinguish among typicality inclusions by means of their probabilities: given two typical properties of a given concept, one can formalize the fact that the probability of having exceptional elements is different for the two properties/inclusions. Prob- abilities of exceptions are then used in order to reason about plausible scenarios, obtained by selecting only some typicality assumptions and whose probabilities belong to a given and fixed range. PEAR is implemented in Python and it pro- vides a decision procedure for reasoning in the Description Logic ALC+T^P_R, which turns out to be inExpTimeas in the underlying standardALC.

An extension of DLs of typicality with probabilities is presented in [17], however probabilities are intended as degrees of belief as in the DISPONTE semantics [15].

Fig. 2.Some pictures of PEAR. On the left, it can be observed the list of generated sce- narios. On the right, PEAR shows the probability of the queryInstagramUser(thomas).

PEAR represents a preliminary attempt to implement reasoning services for the logicALC+T^P_R. We plan to provide a more mature version, by investigating the applicability of techniques introduced in [18, 19] in order to improve the efficiency of the system.

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